The most common technique for removing the effect of fading in the received signal of a wireless radio channel is to mitigate the effect of fading at the transmitter by controlling the transmitter's power using precoding. Thus, if the channel coefficients are known at the transmitter then the transmitter can change the transmitted signal power to overcome the effect of the channel fading at the receiver. There are two main problems with this solution. The first is increasing the dynamic range of transmitter by this solution and the second is that the transmitter does not have any knowledge of the channel as known by the receiver (except time division duplex systems, where the transmitter receives power from another known transmitter over the same channel).
Another approach against fading effects was disclosed by Alamouti et al (U.S. Pat. No. 6,185,258), titled “Transmitter Diversity Technique for Wireless Communication,” the entire contents of which are hereby expressly incorporated herein. In this disclosure, an arrangement with two transmit antennas can be realized that provides diversity with bandwidth efficiency, easy decoding at the receiver (merely linear processing), and performance that is the same as the performance of maximum ratio combining arrangements. This approach also uses maximum ratio combiner as linear maximum likelihood receiver. The rate of these codes which are named orthogonal codes is less than one but have good performance in the fading channels than 2 antennas but this structure suffers from low transitions rate. However, for more than two antennas, the orthogonal code's rate can not exceed 3/4 see also Wang, H. and Xia, X.-G. Upper bounds of rates of space-time block codes from complex orthogonal designs. IEEE Trans. on Information Theory, 49(10): October 2003, 2788-96, the entire contents of which are hereby expressly incorporated herein.
Consider a space time code X, uses M antenna for transmission of M symbols in M time slot such as:
  C  =      [                                        c            11                                                c            12                                    …                          …                                      c                          1              ⁢              M                                                                        c            21                                                c            22                                    …                          …                                      c                          2              ⁢              M                                                            ⋮                          ⋮                          ⋱                                                                          ⋮                                      ⋮                          ⋮                                                                          ⋱                          ⋮                                                  c                          M              ⁢                                                          ⁢              1                                                            c                          M              ⁢                                                          ⁢              2                                                …                          …                                      c            MM                                ]  
where Cij is the transmitted symbol from ith antenna in jth time slot.
In “Space-time codes for high data rate wireless communication: performance analysis and code construction.” IEEE Trans. on Information Theory, 44: March 1998, 744-65, the entire contents of which are hereby expressly incorporated herein. Tarokh et al proposed some criteria for design of space-time codes. For any two codewords C≠C′, the rank criterion suggests that the error matrix D(C,C′)=C−C′ has to be full rank. The proposed structure provides space-time codes with rate equal to 1 and 2 for arbitrary number of transmit antennas. Therefore, this structure has higher transitions rate than the mentioned methods. However, since the proposed code does not have an orthogonal structure in the receiver side, a linear optimum receiver effectively can not be implemented.
Therefore, there is a need for a non-orthogonal structure, which use a sphere decoder for optimum decoding. When the transmission rate is 1 and 2 the code does not require QR decomposition and it also benefits from low complexity suboptimum decoders.